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Properties

Label 18T473
Order \(5184\)
n \(18\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

Degree $n$ :  $18$
Transitive number $t$ :  $473$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,13,3,7,16,5,10,18)(2,12,14,4,8,15,6,9,17), (1,15,7)(2,16,8)(3,17,9,4,18,10)(5,13,12,6,14,11)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
27:  $C_3^2:C_3$
81:  $C_3 \wr C_3 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 9: $C_3 \wr C_3 $

Low degree siblings

12T265

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 7, 8)( 9,10)(13,14)(17,18)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $27$ $2$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $2$ $(13,14)(15,16)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,16,18)(14,15,17)$
$ 3, 3, 3, 3, 3, 3 $ $64$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(13,18,16)(14,17,15)$
$ 9, 9 $ $576$ $9$ $( 1,11,13, 3, 7,16, 5,10,18)( 2,12,14, 4, 8,15, 6, 9,17)$
$ 9, 9 $ $576$ $9$ $( 1,13, 7, 5,18,11, 3,16,10)( 2,14, 8, 6,17,12, 4,15, 9)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $48$ $3$ $( 1, 6, 3)( 2, 5, 4)(13,16,18)(14,15,17)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $144$ $6$ $( 1, 6, 3)( 2, 5, 4)( 7, 8)( 9,10)(13,16,17)(14,15,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $48$ $3$ $( 1, 3, 6)( 2, 4, 5)(13,18,16)(14,17,15)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $144$ $6$ $( 1, 3, 6)( 2, 4, 5)( 7, 8)( 9,10)(13,17,15)(14,18,16)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $(13,17,16)(14,18,15)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 7, 8)( 9,10)(13,18,15)(14,17,16)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 1, 2)( 3, 4)(13,18,16)(14,17,15)$
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $108$ $6$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,17,15)(14,18,16)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $144$ $6$ $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,14)(17,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $48$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)$
$ 3, 3, 3, 3, 3, 3 $ $192$ $3$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(13,15,17)(14,16,18)$
$ 6, 6, 3, 3 $ $432$ $6$ $( 1,11,13, 2,12,14)( 3, 7,16)( 4, 8,15)( 5,10,18, 6, 9,17)$
$ 3, 3, 3, 3, 3, 3 $ $144$ $3$ $( 1,11,13)( 2,12,14)( 3, 7,15)( 4, 8,16)( 5,10,17)( 6, 9,18)$
$ 9, 9 $ $576$ $9$ $( 1,13,12, 4,15, 8, 6,17,10)( 2,14,11, 3,16, 7, 5,18, 9)$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $3$ $(13,16,17)(14,15,18)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 7, 8)( 9,10)(13,16,18)(14,15,17)$
$ 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $6$ $( 1, 2)( 3, 4)(13,15,17)(14,16,18)$
$ 3, 3, 2, 2, 2, 2, 1, 1, 1, 1 $ $108$ $6$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,15,18)(14,16,17)$
$ 3, 3, 3, 3, 3, 3 $ $192$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,10,11)( 8, 9,12)(13,18,15)(14,17,16)$
$ 3, 3, 3, 3, 2, 2, 1, 1 $ $144$ $6$ $( 1, 5, 3)( 2, 6, 4)( 7,11,10)( 8,12, 9)(15,16)(17,18)$
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $48$ $3$ $( 1, 5, 4)( 2, 6, 3)( 7,11,10)( 8,12, 9)$
$ 9, 9 $ $576$ $9$ $( 1,11,13, 5,10,18, 4, 8,15)( 2,12,14, 6, 9,17, 3, 7,16)$
$ 6, 6, 3, 3 $ $432$ $6$ $( 1,13,10)( 2,14, 9)( 3,16,12, 4,15,11)( 5,18, 8, 6,17, 7)$
$ 3, 3, 3, 3, 3, 3 $ $144$ $3$ $( 1,14, 9)( 2,13,10)( 3,16,12)( 4,15,11)( 5,17, 7)( 6,18, 8)$

Group invariants

Order:  $5184=2^{6} \cdot 3^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.